Floating (or sinking) Bodies. Robert Finn

Transcription

1 Floating (or sinking) Bodies Robert Finn

2 Place rigid body B of density ρ into bath of fluid of density ρ 0 in vertical gravity field g, assuming no interaction of adjacent materials. Then B floats if ρ < ρ 0 and sinks if ρ > ρ 0 (Archimedes Law). 2 g B

3 Place rigid body B of density ρ into bath of fluid of density ρ 0 in vertical gravity field g, assuming no interaction of adjacent materials. Then B floats if ρ < ρ 0 and sinks if ρ > ρ 0 (Archimedes Law). 3 g B In fact adjacent materials do interact, and the change can be dramatic, with very different behavior. Aristoteles already knew 350 BC that heavy bodies can float on water. Now known that if B floats then orientation and height of B adjusts, and outer free surface changes, to meet B in contact angle γ, depending on materials.! B!

4 Nothing further for almost 2000 years, when Galileo questioned Aristoteles reasoning in his Discorsi, about Then 200 years later Laplace took up the topic, using the then new concept of surface tension, and the relatively new magic weapon of the Calculus, obtaining some remarkable although not yet complete mathematical results. 4

5 Nothing further for almost 2000 years, when Galileo questioned Aristoteles reasoning in his Discorsi, about Then 200 years later Laplace took up the topic, using the then new concept of surface tension, and the relatively new magic weapon of the Calculus, obtaining some remarkable although not yet complete mathematical results. 5 Then another quietus for almost 200 years, till some engineering literature, and a single mathematics paper, by Raphaël, di Meglio, Berger, and Calabi in Idealized case, g = 0, two dimensions (horizontal infinite cylinder), convex section B, fluid horizontal and at same height on both sides, meeting B with contact angle γ. B!! S

6 6 Nothing further for almost 2000 years, when Galileo questioned Aristoteles reasoning in his Discorsi, about Then 200 years later Laplace took up the topic, using the then new concept of surface tension, and the relatively new magic weapon of the Calculus, obtaining some remarkable although not yet complete mathematical results. Then another quietus for almost 200 years, till some engineering literature, and a single mathematics paper, by Raphaël, di Meglio, Berger, and Calabi in Idealized case, g = 0, two dimensions (horizontal infinite cylinder), convex section B, fluid horizontal and at same height on both sides, meeting B with contact angle γ. B!! S Theorem: There exist at least four positions of B yielding prescribed γ.

7 Theorem: There exist at least four positions yielding prescribed γ. Proof: From Four vertex theorem of differential geometry 7 Example: Ellipse yields exactly four!!

8 Theorem: There exist at least four positions yielding prescribed γ. Example: Ellipse yields exactly four 8!! Question: What are the shapes that will float in every orientation?

9 Theorem: There exist at least four positions yielding prescribed γ. Example: Ellipse yields exactly four 9!! Question: What are the shapes that will float in every orientation? Conjecture: Must be circle.

10 Theorem: There exist at least four positions yielding prescribed γ. Example: Ellipse yields exactly four 10!! Question: What are the shapes that will float in every orientation? Conjecture: Must be circle. Counterexample: If γ = π/2 then any curve of constant width will work!

11 11! C d(!) d(ν ) = width in direction ν Defn. constant width if d(ν ) independent of ν.

12 12! C d(!) d(ν ) = width in direction ν Defn. constant width if d(ν ) independent of ν. N.B. There is some indication that the disaster on the spacecraft Challenger happened at least in part because the engineers assumed it self-evident that all curves of constant width are circles. The engineers determined the circularity of the port to a fuel module by measuring its width in three different directions.

13 13 Example due to Rabinowitz (1997): x = 9cos!+2cos2! cos4! y= 9sin! 2sin2! sin4! Procedure yields non-countable family of counterexamples.

14 I gave a different construction, by modifying ellipses, and obtained a different non-countable family, again with γ = π/2. I tried for some time to construct examples for other γ, without success. 14

15 I gave a different construction, by modifying ellipses, and obtained a different non-countable family, again with γ = π/2. I tried for some time to construct examples for other γ, without success. As it turned out, I would have done much better had I known modern billiards theory. Every planar body B that floats in every orientation with angle γ can be viewed as a billiard table with an invariant curve at angle π γ. 15

16 I gave a different construction, by modifying ellipses, and obtained a different non-countable family, again with γ = π/2. I tried for some time to construct examples for other γ, without success. As it turned out, I would have done much better had I known modern billiards theory. Every planar body B that floats in every orientation with angle γ can be viewed as a billiard table with an invariant curve at angle π γ. 16 γ. > π γ B π γ π γ γ

17 I gave a different construction, by modifying ellipses, and obtained a different non-countable family, again with γ = π/2. I tried for some time to construct examples for other γ, without success. As it turned out, I would have done much better had I known modern billiards theory. Every planar body B that floats in every orientation with angle γ can be viewed as a billiard table with an invariant curve at angle π γ. 17 γ. > π γ B π γ π γ γ There is a theorem by E. Gutkin, that a non-circular billiard table with invariant curve at angle α exists if and only if α satisfies for some n 3. tannα = ntanα

18 I gave a different construction, by modifying ellipses, and obtained a different non-countable family, again with γ = π/2. I tried for some time to construct examples for other γ, without success. As it turned out, I would have done much better had I known modern billiards theory. Every planar body B that floats in every orientation with angle γ can be viewed as a billiard table with an invariant curve at angle π γ. 18 γ. > π γ B π γ π γ γ There is a theorem by E. Gutkin, that a non-circular billiard table with invariant curve at angle α exists if and only if α satisfies for some n 3. tannα = ntanα We conclude that the set of contact angles γ, for which a non-circular body can be found that will float in every orientation with angle γ, is countable and everywhere dense.

19 In 3-D situation very different. The external fluid surface is no longer automatically flat, and for analogous theory that must be added as hypothesis. We label that as neutral equilibrium. 19

20 In 3-D situation very different. The external fluid surface is no longer automatically flat, and for analogous theory that must be added as hypothesis. We label that as neutral equilibrium. Theorem (joint with Mattie Sloss): If for some γ (0,π) a strictly convex body B will float in neutral equilibrium in every orientation, then B is a metric ball. 20

21 In 3-D situation very different. The external fluid surface is no longer automatically flat, and for analogous theory that must be added as hypothesis. We label that as neutral equilibrium. Theorem (joint with Mattie Sloss): If for some γ (0,π) a strictly convex body B will float in neutral equilibrium in every orientation, then B is a metric ball. Proof: Based on Joachimstal s Theorem in differential geometry: If two surfaces meet at a constant angle, and if the intersection curve is a curvature line on one of the surfaces, then it is a curvature line on both. 21

22 All of above was formal geometrical, and assumed floating in zero gravity, giving no information toward problem suggested by Arisoteles, as to conditions under which heavy bodies can float. We attack that problem with principle of virtual work, by minimizing energy. 22

23 All of above was formal geometrical, and assumed floating in zero gravity, giving no information toward problem suggested by Arisoteles, as to conditions under which heavy bodies can float. We attack that problem with principle of virtual work, by minimizing energy. Some of energy is gravitational, E g = work against gravity needed to create a given configuration from a prescribed reference state. 23

24 All of above was formal geometrical, and assumed floating in zero gravity, giving no information toward problem suggested by Arisoteles, as to conditions under which heavy bodies can float. We attack that problem with principle of virtual work, by minimizing energy. Some of energy is gravitational, E g = work against gravity needed to create a given configuration from a prescribed reference state. There is also a surface energy E S = work needed to create surface interface, due to unequal attractions E S = es, where e = energy per unit area, depending only on materials. For fluid/fluid interface, e σ = surface tension.

25 We seek to minimize the total energy, E = E S + E g. 25 Let e 0, e 1, e 2 be surface energy densities on outer, upper, lower interfaces. Start again with 2-D case, with g = 0. e 0 B e 1 S e2

26 We seek to minimize the total energy, E = E S + E g. 26 Let e 0, e 1, e 2 be surface energy densities on outer, upper, lower interfaces. Start again with 2-D case, with g = 0. e 0 B e 1 S e2 Theorem: If B is strictly convex, and if g = 0, then a strictly minimizing partly wetting configuration exists if and only if e 1 e 2 e 0 <1. Theorem: Whether or not g = 0, if e 1 e 2 < e 0 then at a local minimum the contact angle γ is determined by e 1 = e 2 + e 0 cosγ. Example: Two of the ellipse positions minimize, the others are unstable.

27 If g 0 basic changes to previous considerations. If the tank infinitely deep, a fully submerged B must sink if its density ρ > ρ 0. Thus there can be no global minimum for the energy E. But conceivably a partially wetted position can occur, with a local minimum for E. 27

28 If g 0 basic changes to previous considerations. If the tank infinitely deep, a fully submerged B must sink if its density ρ > ρ 0. Thus there can be no global minimum for the energy E. But conceivably a partially wetted position can occur, with a local minimum for E. Behavior controlled by limitations on how much the outer surface S can bend. In 2-D can determine the most general S explicitly, as solution of ( sin!) x = "u Here ψ = inclination angle, κ = ρg/σ. We get x = x ! " sin" # d" 0 1 sin" 28 u = 2! ( 1 sin" ) with ϕ = (π/2) ψ.

29 If g 0 basic changes to previous considerations. If the tank infinitely deep, a fully submerged B must sink if its density ρ > ρ 0. Thus there can be no global minimum for the energy E. But conceivably a partially wetted position can occur, with a local minimum for E. Behavior controlled by limitations on how much the outer surface S can bend. In 2-D can determine the most general S explicitly, as solution of ( sin!) x = "u Here ψ = inclination angle, κ = ρg/σ. We get x = x ! " sin" # d" 0 1 sin" 29 u = 2! ( 1 sin" ) with ϕ = (π/2) ψ. Note that the height change from infinity cannot exceed 2/ κ. Thus the position of any floating body is localized.

30 Note that the height change from infinity cannot exceed 2/ κ. Thus the position of any floating body is localized. This limits the effects of buoyancy forces from within the liquid. We conclude: Let h c be height of centroid of B, and d = dia{b}. If h c > 2 e 0 / (! 2! 1 )g + d then B disjoint from S, hence E g < A + B e 0 when floating, where A, B depend only on shape of body and gravitational quantities, not on surface energies. So from bottom to top of possible positions δe g < 2 (A + B e 0 ). 30

31 We conclude: Let h c be height of centroid of B, and d = dia{b}. If h c > 2 e 0 / (! 2! 1 )g + d then B disjoint from S, hence E g < A + B e 0 when floating, where A, B depend only on shape of body and gravitational quantities, not on surface energies. So from bottom to top of possible positions δe g < 2 (A + B e 0 ). Choose e 0 big enough that 2 (A + B e 0 ) < s B e 0. Then choose e 1, e 2 so that δe S = (e 2 e 1 ) s B < 2 (A + B e 0 ). Then δe S < δe g δe < 0 local minimum. 31 Further e 2 e 1 e 0 > 2 (A + B!e 0 ) s B e 0 The right side < 1 we can scale e 1, e 2 to get 1 > e 2 e 1 e 0 > 2 (A + B!e 0 ) s B e 0 and the minimizing configuration achieved with e 1 = e 2 + e 0 cosγ. N.B. This result requires γ > π/2.

32 32 E E S = e 2 s B E + E E S + E g + = e 1s B h c + h c h c E g

33 What happens in 3-D? More complicated since many more possible configurations for S. Restrict attention to the family of symmetric solns: 33 Theorem (joint with Tom Vogel): Any floating body must intersect the closure of the set of symmetric extremals, for every choice of the center of symmetry.

34 What happens in 3-D? More complicated since many more possible configurations for S. Restrict attention to the family of symmetric solns: 34 Theorem (joint with Tom Vogel): Any floating body must intersect the closure of the set of symmetric extremals. for every choice of the center of symmetry. Thus, the red configuration is not possible, regardless of the density or surface properties of the body. If a small body floats it must be closer to the surface than must a large one of the same shape.

35 What happens in 3-D? More complicated since many more possible configurations for S. Restrict attention to the family of symmetric solns: 35 Theorem (joint with Tom Vogel): Any floating body must intersect the closure of the set of symmetric extremals. for every choice of the center of symmetry. Thus, the red configuration is not possible, regardless of the density or surface properties of the body. If a small body floats it must be closer to the surface than must a large one of the same shape. Theorem: Any body of given density and with γ > π/2 can be made to float by scaling its size down sufficiently.

36 36 Theorem (2 or 3 D): If one holds B rigidly far enough above or below the rest level at infinity, then B will be disjoint from S. If one then moves B rigidly vertically from top to bottom (or reverse), the motion of S will be discontinuous. Proof: The height of S is given by a solution of divtu =!u,!!!!!tu " #u,!!!! > #u The only global solution for S that vanishes at infinity is the flat one u 0.

37 37 Historical comments Apparently the first recorded discussion of capillarity phenomena is due to Aristoteles c. 350 B.C, and addressed exactly the question that has since been most ignored. He wrote: A large flat body, even of heavy material, will float on water, but a long thin one such as a needle will always sink.

38 38 Historical comments Apparently the first recorded discussion of capillarity phenomena is due to Aristoteles c. 350 B.C, and addressed exactly the question that has since been most ignored. He wrote: A large flat body, even of heavy material, will float on water, but a long thin one such as a needle will always sink. 1. Above theory when applied to thin circular horizontal disk of areal density ρ 0 yields that if! 2 0 > 2 (! 2! 1 ) / "g, then if radius large enough it must sink.

39 39 Historical comments Apparently the first recorded discussion of capillarity phenomena is due to Aristoteles c. 350 B.C, and addressed exactly the question that has since been most ignored. He wrote: A large flat body, even of heavy material, will float on water, but a long thin one such as a needle will always sink. 1. Above theory when applied to thin circular horizontal disk of areal density ρ0 yields that if!20 > 2 (!2!1 ) / "g, then if radius large enough it must sink. 2. Experimental response:

40 Formal calculations suggest that a thin metal needle with γ as small as 16 deg can be made to float in equilibrium on water. Thus my own criterion, although perhaps the first sufficient one known, may not be necessary, as it requires γ > π/2. Conceivably it is necessary for stable equilibria, at this point I have no idea. In a somewhat other direction, the theory predicts an absolute maximum diameter for which a needle of given material can float in the earth s gravity field, regardless of contact angle. The result indicates that a typical small paper clip can be made to float, but that a large paper clip will always sink. I could confirm the prediction with a kitchen sink experiment. This may be where Aristoteles went wrong; a needle in his time may have had a much larger diameter than the typical modern household product that I used for my picture. 40

41 41 The end. Thank you for listening!